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I read some time ago about terraria (singular: terrarium), which are basically small ecosystems of plants, isolated from the outside world (they get just energy from outside, not water or oxygen). The question is how big should such a terrarium be to support a human, assuming it has an infinite energy source and today's technology?

Given the technology available, it is possible to create an artificial environment, where the temperature and humidity is controlled. I believe even the composition of the air can be controlled to some extent, by having some air separation plants and adding some gases to increase their concentration. Also, the water is not a problem, as there are artificial environments where water is recycled (like the International Space Station).

So 2 problems remain: oxygen and food. I think the main source of food should be plants. A nice side-effect of growing plants is that they generate oxygen. Actually, if they produce enough food, they produce enough oxygen. But what kind of plants and how much plants should be harvested in order to produce enough food and oxygen for a man?

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    $\begingroup$ As one lower bound, Biosphere 2 proved very difficult to balance. $\endgroup$ – Cort Ammon Jan 2 '15 at 3:34
  • $\begingroup$ Self-sustained for how long? The answer will be different for years and for millions of years. $\endgroup$ – Irigi Jan 2 '15 at 7:59
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    $\begingroup$ Let's say self sustained for a life time (50-100 years). And let's assume the equipment won't break, so new parts don't need to be produced. So, the main concern is creating an artificial biosphere that can support a human. $\endgroup$ – Paul92 Jan 2 '15 at 13:14
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    $\begingroup$ It's not an answer but a one person terrarium would likely drive the vast majority of humanity insane. $\endgroup$ – bowlturner Jan 2 '15 at 14:17
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    $\begingroup$ One big question is energy. Everything else can be solved with energy, but where does that come from? Is this on an earth-like planet with sunlight? do you have to relay on batteries? I can manage everything in less space if i'm tapping a huge stored energy source then if I have to spread it out to get all my energy from sunlight, which is limited energy / ft. also as bowlturner said I would be insane within a month or two (well, I'M already insane, but that's a different issue). $\endgroup$ – dsollen Jan 2 '15 at 14:33
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Problem 1: Food

This is the major contributing factor to the size of the biosphere. Humans need to consume around 2,250 calories per day (averaged between men and women) to sustain themselves, and more if they are doing physical work. 2,250 cal = 10,460 joules (or 10.46kJ), but that's not very useful. More useful: 100g of potato (in any form) = 77cal. One medium potato (defined by Google via USDA as 213g) contains 163 calories. If you live off potatoes alone, you'll need:

$$ 2,250 \div 163 = 13.80 \approx 14\text{ potatoes/day} $$ $$ = 14 \times 365 = 5110 \text{ potatoes/year} $$

That's quite a lot of potatoes. As you can see here, potatoes can be planted March-May and harvested Jun-Oct. You could always plant several crops to be harvested at different times.

The same website as linked above says potatoes should be planted around 30cm apart. Assuming you follow this guide, then you'll need:

$$ 5110 \times 30 = 153300\text{cm} = 1533.00\text{m} $$

for all your potatoes in one row. However, you could plant in multiple rows to save space:

$$ 1533 \div 50 = 30.66 \text{m length} $$ $$ 50 \times 0.3\text{m} = 15\text{m width} $$ $$ 30.66 \times 15 = 459.9\text{m}^{2} $$

So, to survive on potatoes for one year, you will need 460 square metres of space in the biosphere. However, take into account that

  • You won't want to survive on potatoes for a year, so you need more land for other foods;
  • If you do just stick with potatoes, you can't plant them in the same place in consecutive years, so double the land required (920 square metres)
  • Potatoes may not be the best plant. That said, at some quick calculations, carrots would require around 140 square metres less land for the same crop. However, carrots have less calorific content per unit (41 per medium carrot), so you would need more. Growing wheat, while more space efficient, requires work after it's harvested an other ingredients to bring it to an edible state.

Problem 2: Oxygen

I see two options. You can either filter outside air or grow lots of plants.
Filter outside air
If you're assuming today's tech, you need some costly and space-inefficient filtering units. If you have advanced tech, then the walls of your biosphere can be filters, but I'll stick to today's tech. This site has a very good explanation of the oxygen requirements of an average human. To summarise:

8.3ft2 of air per adult for 5 hours (with an 8ft ceiling)

However, this is a bare minimum. If the size is much less, you won't last 5 hours. The FEMA recommendation is 10ft2 per adult for 5 hours. Given that air is 21% oxygen, this comes to $8.3 \times 0.21 = 1.74$ square feet for 5 hours. Using more appropriate numbers, this is $1.74 \times 8 = 13.92$ cubic feet for 5 hours, and $13.92 \div 5 = 2.78$ cubic feet per hour. That sounds like a fairly reasonable number: what output would you need?

$$ (2.78 \div 21) \times 100 = 13.23 $$

This line shows that 2.78 cubic feet per hour of oxygen equates to 13.23 cubic feet per hour of normal air.

$$ (13.23 \div 60) \div 60 = 0.003675 $$

This is the amount of air that your concentrator needs to take in per second to keep you alive. That's a perfectly good number: given that many home fans can move around 2 cubic feet of air per second, a commercial fan in a concentrator will have no trouble. A unit capable of doing this takes up less than 1 square metre.

Grow lots of plants
According to NASA, the average human needs 0.84kg of oxygen each day to survive. This can be produced by any size plants, but the space requirement changes depending. For example, a mature broadleaf tree such as an oak produces 10-15kg of oxygen per day, plenty to support a human. However, it can easily take up 20 square metres of space and can reach up to 30m high. Smaller plants may be easier: since they produce less oxygen, you can use a more exact number to produce closer to the right amount of oxygen per day, saving space. However, keep in mind that an exercising human uses much more oxygen - up to 7kg per day.

The really easy way is to just pull in air from outside. However, this may not be possible, so your best bet is probably the filter, as it takes up the least space.

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    $\begingroup$ +1 for defining human caloric requirements in potatoes. To feed 1000 humans in a day, you need a kilotater, 1,000,000 humans, megatater. $\endgroup$ – Sidney Jan 2 '15 at 19:10
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    $\begingroup$ While I like part 1 of the answer (and the numbers seem to fit a quick search on the internet, asuming good yields of your plant), but: using air from the outside really does not qualify as self-sustaining. The number (200 plants) is really difficult to be useful as that could be any size. I found: human oxygen consumption 0.5 to 2 kg/day. A sufficiently aged, medium sized broadleaf tree produces 10 to 15 kg/day (again highly depending on a multitude of factors, size, humidity, solar insolation and so on). $\endgroup$ – Ghanima Jan 2 '15 at 19:44
  • $\begingroup$ Nice answer. However, my initial question assumed you can't get anything from outside, except for energy in the form of electricity. So filtering air is not an option. However, if you grow enough plants to make enough food to eat, shouldn't they generate the required amount of oxygen? They absorb all the CO2 you generate by breathing. Also, while your computations are very interesting, is potato the best plant? Aren't there any other plants with faster growth and that can be consumed in a higher proportion than a potato? $\endgroup$ – Paul92 Jan 2 '15 at 19:46
  • $\begingroup$ @Paul92 Have to admit to wriying that in a bit of a rush, so I havn't looked at other plants. I'll update this tomorrow. $\endgroup$ – ArtOfCode Jan 2 '15 at 21:26
  • $\begingroup$ @Ghanima I was assuming small plants. As I say, I'll update tomorrow. $\endgroup$ – ArtOfCode Jan 3 '15 at 0:37
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Biosphere 2 ran into some problems because they didn't take into account the heating and cooling effect on the air. You need a big bladder to handle the pressure differences if you're not heating and cooling the thing with machinery (which will likely throw off your plants' cycles)

Oxygen is not a problem, if you've got artificial light and water. Just pump your CO2 thru, and there's enough algae spores/whatever in the air (unless you treated the air prior to pumping it in) to colonize that water and make you some O2

In fact, humans can eat algae. But, yeech (ie: dietary boredom can lead to refusal to eat/lack of interest in food/lack of will to live. related: dysphagia). Biggest problem is going to be getting the correct amino acids (humans don't produce all they need), and trace elements (which may, or may not be doable with micro-plants). You'll need to develop some things to cycle those. But yeah, put some plants together. If you've got enough machinery, and a lot less care of the human - you could probably put together something the size of the room (total-size), and keep them in a gel pod ala Matrix and pump the fluids and gases thru their meat-bag, recycling them with plants and sun/artificial lighting - or with vast amounts of energy and chemical (nano) machinery (ie: the rest of a 10x10x10). Best bet, hook them up to an artificial sensory environment, so they don't know how isolated and un-free they are - and how limited their diet is.

Aquaponics will get you delicious fish (maybe shrimp) and plants to eat, and compost your manure (after sterilizing; no e.coli contamination) and toss it right back into the system.

If you don't want to take your human's brain offline, you're going to need some VR, and an infinite walker. And maybe some serious space for them to wander around. Still likely to get claustrophilia and or caged-tiger syndrome when you try and take them out later.

Why do you need your humans put on hold? Why not stasis? Or cryogenics? Or sperm-cell banks?

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    $\begingroup$ Unfortunately, no one can be told what the Matrix is. $\endgroup$ – Ghanima Jan 3 '15 at 10:03
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Look at the size of Biosphere 2. It failed to support human life indefinitely, but they learned a lot from it. Look over that and find issues that are one-time planning/building mistakes (e.g. the concrete curing uses oxygen), things that could be done better once you know how, and (the interesting part) what fundamentally didn't work due to the scale?

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